The modern model of the Solar System is basically that of Isaac Newton. His
laws of motion and universal gravitation accurately predict the motions of all
the planets, asteroids, comets, natural and artificial satellites in the solar
system to very high precision. This is very powerful evidence that the model
is correct and the Earth, therefore, MUST go around the Sun. But is there any
more direct evidence for motion of the Earth?

Yes, there is. I'll mention two pieces of evidence, abberation of starlight and
stellar parallax. These share some interesting history because abberation of
starlight was discovered when experiments were done to detect stellar
parallax. So, what are these things?

If the Earth goes around the Sun then our postion changes throughout the
year. We now know that this motion causes
Earth to move, back and forth, about 186 million miles each year. If we
measure the position of stars in the sky, our changing viewpoint should cause
their positions to shift. How much is this shift? When the attempt to measure this
was first
made by James Bradley and
Samuel Molyneux, in 1725-1727, it was assumed that all stars were the same brightness
as the sun. The star that was chosen, Gamma Draconis, is pretty bright. About
the same as the stars in the Big Dipper. Assuming it was as bright as the sun
it must be really far away. It worked out to a total shift of 0.3 arcseconds.

This was just within the expected accuracy of the telescope used. The actual
experimental result was more than 10 times bigger
and the shift happened at the wrong time of the year! The effect was the same
for all of the stars in the same part of the sky. In fact it depended only on
the star's distance from the path of the Sun in the sky. This was seen as the
result of the finite speed of light. The effect is called the abberation of
starlight.

What about stellar parallax? It turned out the stars were further away than
people had supposed and more sensitive instruments were needed to measure
parallax. Finally, in 1838, Friedrich Bessel
measured
the parallax of a star.